Addition-chain defect truncation order-type conjecture

Let $\ell(n)$ denote the addition-chain length of the natural number $n$, and define the addition-chain defect by

$$\delta^{\ell}(n):=\ell(n)-\log_2 n.$$

Let

$$\mathscr{D}^{\ell}:=\{\delta^{\ell}(n):n\in\mathbb{N}\}.$$

Addition-chain defect order-type conjecture. For each whole number $k$, the set $\mathscr{D}^{\ell}\cap[0,k]$ has order type $\omega^k$. This conjecture concerns the fine structure of the well-ordered set of addition-chain defects, whose overall order type is already known to be $\omega^\omega$.